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Slope and y-intercept · 5 marks

Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’.

  1. (i) y = 4x, y = 2x, y = x
  2. (ii) y = − 6x, y = − 3x, y = − x
  3. (iii) y = 5x, y = −5x
  4. (iv) y = 3x − 1, y = 3x, y = 3x + 1
  5. (v) y = −2x − 3, y = −2x, y = 2x + 3
Answer: (i) All through the origin (b = 0); larger a = steeper. (ii) All through the origin, falling; larger size of a = steeper. (iii) Equally steep, one rising and one falling, mirror images in the y-axis. (iv) Parallel (same a = 3), cutting the y-axis at −1, 0, 1. (v) y = −2x − 3 and y = −2x are parallel; y = 2x + 3 rises and meets y = −2x − 3 on the x-axis at (−1.5, 0).

Step-by-step solution

Idea: In y = ax + b: a is the slope (its sign gives the direction, its size the steepness) and b is the y-intercept (where the line cuts the y-axis). Two points, such as x = 0 and x = 1, are enough to draw each line.

(i) y = 4x, y = 2x, y = x

  1. Points: (0, 0) and (1, 4); (0, 0) and (1, 2); (0, 0) and (1, 1).xy−4−3−2−1123−5−4−3−2−112340y = 4xy = 2xy = x½ mark
  2. Role of a and b: b = 0 for all, so all pass through the origin. a = 4, 2, 1 are positive, so the lines rise; the larger a, the steeper the line (y = 4x steepest).½ mark
All through the origin (b = 0); y = 4x steepest, y = x least steep.

(ii) y = − 6x, y = − 3x, y = − x

  1. Points: (0, 0) and (1, −6); (0, 0) and (1, −3); (0, 0) and (1, −1).xy−4−3−2−1123−5−4−3−2−112340y = −6xy = −3xy = −x½ mark
  2. Role of a and b: b = 0, so all pass through the origin. a is negative, so the lines fall from left to right; the larger the size of a, the steeper (y = −6x steepest).½ mark
All through the origin and falling; y = −6x steepest, y = −x least steep.

(iii) y = 5x, y = −5x

  1. Points: (0, 0) and (1, 5); (0, 0) and (1, −5).xy−4−3−2−1123−5−4−3−2−112340y = 5xy = −5x½ mark
  2. Role of a and b: b = 0, so both pass through the origin. The slopes 5 and −5 have the same size, so the lines are equally steep, but y = 5x rises and y = −5x falls. They are mirror images in the y-axis.½ mark
Both through the origin, equally steep; one rises, one falls.

(iv) y = 3x − 1, y = 3x, y = 3x + 1

  1. Points: (0, −1) and (1, 2); (0, 0) and (1, 3); (0, 1) and (1, 4).xy−4−3−2−1123−5−4−3−2−112340y = 3x + 1y = 3xy = 3x − 1½ mark
  2. Role of a and b: a = 3 for all, so the lines are equally steep and parallel. b = −1, 0, 1 shifts the line: they cut the y-axis at (0, −1), (0, 0), (0, 1).½ mark
Parallel lines (slope 3) with y-intercepts −1, 0 and 1.

(v) y = −2x − 3, y = −2x, y = 2x + 3

  1. Points: (0, −3) and (−1.5, 0); (0, 0) and (1, −2); (0, 3) and (−1.5, 0).xy−5−4−3−2−11234−6−5−4−3−2−1123450y = −2x − 3y = −2xy = 2x + 3½ mark
  2. Role of a and b: y = −2x − 3 and y = −2x have the same a = −2, so they are parallel and fall to the right; b = −3 shifts the first one 3 units down. y = 2x + 3 has a = +2, so it rises and is not parallel to them; it is the mirror image of y = −2x − 3 in the x-axis, and the two meet at (−1.5, 0). (As printed, the third line is 2x + 3; if −2x + 3 was meant, all three would be parallel with y-intercepts −3, 0, 3.)½ mark
y = −2x − 3 and y = −2x are parallel; y = 2x + 3 rises and crosses y = −2x − 3 at (−1.5, 0).
a decides the direction and steepness (positive rises, negative falls, larger size is steeper); b decides where the line cuts the y-axis. Lines with the same a are parallel ((iv), and the first two in (v)); lines with b = 0 pass through the origin ((i)−(iii)).

Check: (v): solving −2x − 3 = 2x + 3 gives x = −1.5, y = 0, so the two lines meet at (−1.5, 0) on the x-axis ✓.

Answer to write in the exam

(i)

y = 4x: (0, 0), (1, 4); y = 2x: (0, 0), (1, 2); y = x: (0, 0), (1, 1)

Graph drawn

∴ b = 0: all pass through the origin; larger a > 0 ⇒ steeper rising line

(ii)

y = −6x: (0, 0), (1, −6); y = −3x: (0, 0), (1, −3); y = −x: (0, 0), (1, −1)

Graph drawn

∴ b = 0: all pass through the origin; a < 0 ⇒ falling lines; larger size of a ⇒ steeper

(iii)

y = 5x: (0, 0), (1, 5); y = −5x: (0, 0), (1, −5)

Graph drawn

∴ b = 0: both through the origin; a = 5 rises, a = −5 falls; equally steep (mirror images in the y-axis)

(iv)

y = 3x − 1: (0, −1), (1, 2); y = 3x: (0, 0), (1, 3); y = 3x + 1: (0, 1), (1, 4)

Graph drawn

∴ Same a = 3 ⇒ parallel lines; b = −1, 0, 1 ⇒ y-intercepts −1, 0, 1

(v)

y = −2x − 3: (0, −3), (−1.5, 0); y = −2x: (0, 0), (1, −2); y = 2x + 3: (0, 3), (−1.5, 0)

Graph drawn

∴ y = −2x − 3 ∥ y = −2x (same a = −2, y-intercepts −3 and 0); y = 2x + 3 (a = 2) rises and meets y = −2x − 3 at (−1.5, 0)

Common mistakes that cost marks

  • Thinking a larger b makes a line steeper. b only moves the line up or down.
  • Calling y = −6x less steep than y = −x because −6 < −1. Steepness depends on the size, 6 > 1.
  • Reading y = 2x + 3 in (v) as parallel to the other two. Its slope is +2, not −2.

How this can come in the exam

MCQ (1 mark)

Which pair of lines is parallel?

  1. y = 4x and y = −4x
  2. y = 4x + 1 and y = x + 4
  3. y = −4x + 2 and y = −4x − 9
  4. y = 4 and x = 4
Show answer

(C) y = −4x + 2 and y = −4x − 9
Both have slope −4 and different intercepts.

Short answer (3 marks)

Draw y = −x + 2, y = −x and y = −x − 2 on the same axes and write what you notice.

Show answerPoints (0, 2), (2, 0); (0, 0), (1, −1); (0, −2), (−2, 0) (1 mark). Graph (1 mark). They are parallel (slope −1), cutting the y-axis at 2, 0, −2 (1 mark).

Try one yourself

Without drawing, say how the graphs of y = 12x + 4 and y = 12x − 4 are related.

Show answer

Parallel (same slope ½); one cuts the y-axis at (0, 4), the other at (0, −4).

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